If $f$ is a bounded real valued function, then what can be said about convergence of the series $\sum_{t=0}^{\infty} \beta^t f(t)$ with $\beta <1$? Does the limit have to exist, and does it have to be finite?
2026-03-30 14:18:03.1774880283
Does this series have to converge?
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If $f$ is bounded then there exists a constant $M>0$ such that $|f(x)|<M$ for every $x$. This allows us to estimate $$|\sum_{t=0}^\infty\beta^tf(t)|\leq\sum_t|\beta|^t|f(t)|\leq M\sum_t|\beta|^t.$$ If $\beta\in(-1,1)$ then the RHS is finite by the geometric series and thus your series converge. To see that your series doesn't have to converge for other values of $\beta$ just pick $f$ a constant function (bounded) different than $0$ and repeat the estimate above.